Laser Pulse Heating of Spherical Metal Particles

Laser Pulse Heating of Spherical Metal Particles

Michael I. Tribelsky∗

A.N.Nesmeyanov Institute of Organoelement Compounds,Russian Academy of Sciences, Vavilova St. 28, Moscow, 119991, Russia

Moscow State Institute of Radioengineering, Electronics and Automation (Technical University),78 Vernadskiy Ave., Moscow 119454, Russia and

Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Str. 38, Dresden 01187, Germany

Andrey E. MiroshnichenkoNonlinear Physics Centre, Research School of Physical Sciences and Engineering,

Australian National University, Canberra ACT 0200, Australia

Yuri S. KivsharNonlinear Physics Centre, Research School of Physical Sciences and Engineering,

Australian National University, Canberra ACT 0200, Australia

Boris S. Luk’yanchuk†

Data Storage Institute, Agency for Science, Technology and Research, Singapore 117608

Alexei R. KhokhlovA.N.Nesmeyanov Institute of Organoelement Compounds,

Russian Academy of Sciences, Vavilova St. 28, Moscow, 119991, Russia andM.V.Lomonosov Moscow State University, Faculty of Physics,

Lenin Hills, 1, Bldg. 2, Moscow, 119992, Russia

We consider a general problem of laser pulse heating of spherical metal particles with the sizesranging from nanometers to millimeters. We employ the exact Mie solutions of the diffractionproblem and solve heat-transfer equations to determine the maximum temperature at the particlesurface as a function of optical and thermometric parameters of the problem. The main attentionis paid to the case when the thermometric conductivity of the particle is much larger than that ofthe environment, as it is in the case of metal particles in fluids. We show that in this case at anygiven finite duration of the laser pulse the maximum temperature rise as a function of the particlesize reaches an absolute maximum at a certain finite size of the particle, and we suggest simpleapproximate analytical expressions for this dependence which covers the entire range of variationsof the problem parameters and agree well with direct numerical simulations.

PACS numbers: 44.05.+e, 42.62.Be, 82.50.-m

INTRODUCTION

The problem of laser pulse heating of absorbing par-ticles embedded in transparent liquid medium is im-portant for different applications, including stimulationof chemical reactions, laser sintering, selective killingof pathogenical bacteria or cancer cells, etc. A metalnanoparticle excited by laser pulse with the frequencyclose to its plasmon resonance efficiently converts electro-magnetic energy into thermal energy, with dramatic rais-ing of the temperature in the surrounding medium. As aresult, this problem or its substantial parts were a sub-ject of many recent papers in physics, biology, medicineand chemistry (see, e.g., Refs. [1–5] to cite a few).

In spite of many experimental works on the subject, atheoretical understanding of this phenomenon is ratherlimited. In a standard approach, the problem is formu-lated as the study of heat transfer with a source (energyrelease in the particle) obtained as a solution of the cor-

responding diffraction problem. Such a problem does nothave simple exact analytical solutions, and different stud-ies employ either approximate analytical methods (validfor certain limiting cases only) or direct numerical cal-culations with the specified parameters. To the best ofour knowledge, neither general solution of this problemapplicable for a broad range of variations of the prob-lem parameters, nor suitable analytical expressions areavailable in the literature.

On the other hand, for many applications it is highlydesirable to obtain simple analytical expressions whichdescribe the temperature at the surface of the particle Tsin a broad range of the problem parameters, and for theparticle sizes ranging from nanometers to millimeters. Inthe present paper we solve this problem. In particular,we obtain analytical expressions for the maximum tem-perature at the particle surface as a function of opticaland thermometric parameters of the problem. Our re-sults cover a wide range of the particle sizes from a few

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nanometers to millimeters, and they are confirmed bydirect numerical solutions.

The paper is organized as follows. In Sec. II we formu-late the problem and introduce four major spatial scales,namely the particle size, thickness of the skin layer, aswell as the characteristic length of the heat diffusion inthe particle and surrounding medium, which determinea variety of different cases discussed below. Our ana-lytical results and estimates are summarized in Sec. IIIfor twelve different cases which include all possible com-bination of the four spatial scales. Section IV summa-rizes our numerical results which are found to be in agood agreement with the analytical predictions. Finally,Sec. V concludes the paper.

PROBLEM FORMULATION

We consider linear heating of metal particles by a singlelaser pulse with duration τ . We assume that the particlehas the thermometric conductivity χp (typically χp ∼0.1− 1 cm2/s), and it is embedded into a fluid with thethermometric conductivity χf (typically χf ∼ 10−3 −10−2 cm2/s), such that the condition χp � χf alwaysholds, and the heat transfer is limited entirely by theheat diffusion. We neglect convection processes owing tolarge characteristic time required for the convection toarise and develop.

Thus, the heating problem depends on the following setof parameters: the intensity of the laser pulse I, its dura-tion τ , frequency ω, optical and thermometric constantsof the particle and environment. As for the intensity I,owing to the linearity of the problem, the heating rateis just proportional to I, so the results obtained for agiven I may be easily recalculated for any other its valueby a simple scale transformation, see below. As for theother parameters, if the particle is spherical with radiusR and the thickness of the skin layer δ (for metals inthe optical region δ ∼ 10−5 cm), we can define just fourcharacteristic spatial scales of the problem, namely R,δ,√χfτ , and

√χpτ . According to the general principle

of the dimensional analysis, interplay between these fourscales determines the entire variety of heating regimes ofthe particle. All these regimes are discussed below oneby one.

ANALYTICAL RESULTS

Small particles

For small particles, we assume that R� δ, so that theheating occurs in the entire bulk of the particle ratherthan on its surface. Interplay between other scales al-lows to consider several different cases which we analyzebelow.

Case S1: R � δ � √χfτ �√χpτ . In this case

the incident light penetrates into the entire particle andthe energy release should be proportional to the particlevolume, i.e. the absorption cross-section of the particleσabs ∼ R3. Bearing in mind that the dimension of σabsis cm2, it is convenient to write

σabs = αkR3, (1)

where α is a dimensionless quantity, k stands for thewavenumber of the incident light in the environmentalfluid: k = nfω/c. Here nf is purely real refractive indexof the fluid, ω designates the frequency of the incidentlight and c stands for the speed of light in vacuum.

In the simplest case of the Rayleigh scattering compar-ison of Eq. (1) with the well-known expression for σabsfor a spherical small particle [6] provides the followingexpression for α:

α =12πε′′

(ε′ + 2)2

+ ε′′2, (2)

where ε′ and ε′′ stand for the real and imaginary parts ofthe complex relative dielectric permittivity of the particle(ε = εp/n

2f ).

Next, we neglect spatial inhomogeneity of the heatsources, supposing that the volume density of the sourcesinside the particle is a constant equal to σabsI(t)/V ,where I(t) is the power density of the laser pulse (inW/cm2) and V is the particle volume. The ground forthis neglect is that the actual inhomogeneity in the heatsources is rather weak and it results even in weaker tem-perature inhomogeneities, owing to high rate of heattransfer in metals. It allows to replace the actual 3D heattransfer problem by its spherically-symmetric version.

Moreover, in what follows we are interested in esti-mates of the maximal temperature of the particle surface,rather than in its exact calculations. For this reason weemploy the spherically-symmetric problem formulationeven for large particles, when the illuminated part of theparticle obviously has temperature higher than that inshadow.

Taking into account that inequality R� √χfτ meansthe temperature field in the vicinity of the particle isquasi-steady (i.e., the term with the temporal derivativeof T in the heat conduction problem may be neglectedrelative to the terms with the spatial derivatives) we ob-tain that within the framework of the approximationsmade the temperature field is described by the following

3

spherically-symmetric boundary-value problem:

κp1

r2∂

∂r

(r2∂T

∂r

)+

3σI(t)

4πR3= 0, at r < R (3)

1

r2∂

∂r

(r2∂T

∂r

)= 0, at r > R (4)

T (t, R− 0) = T (t, R+ 0), (5)

κp

(∂T

∂r

)R−0

= κf

(∂T

∂r

)R+0

, (6)

T → 0 at r →∞, (7)

where κp and κf are thermal conductivity of the particleand environment, respectively and t in the dependenceT (r, t) plays a role of a parameter. Here and in whatfollows T stands for the temperature rise from the roomtemperature, see Eq. (7).

Integration of Eqs. (3)–(7) yields a parabolic temper-ature profile at r < R and T = Ts(t)R/r at r > R withthe surface temperature of the particle

Ts =σI(t)

4πRκf≡ αR2kI(t)

4πκf. (8)

Note, the temporal dependence of Ts coincides with thatfor I(t), so that the maximal temperature is achieved atthe maximum of the laser pulse.

Case S2: R � √χfτ � δ � √χpτ . The case cor-responds to a quasi-steady field of the temperature bothin the particle and in the fluid. From the point of viewof heat conductivity, the case is identical to (S1), so thatEq. (8) is valid.

Case S3: R � √χfτ �√χpτ � δ. The case is

identical to (S1), and again Eq. (8) is valid.

Large particles

For large particle, we assume that R � δ, so that theenergy release occurs in a thin layer near the particlesurface.

Case L1 δ � R � √χfτ �√χpτ . In this case the

absorption cross-section should be proportional to thesquare of the linear size of the particle, i.e.,

σabs = πR2Qabs, (9)

where Qabs stands for the dimensionless efficiency.Regarding the temperature field, owing to the condi-

tion R � √χfτ �√χpτ it is still quasi-steady. The

difference between the previous case is that now a goodapproximation is the energy release at the surface of theparticle. Then, the temperature field inside the particleshould satisfy the homogeneous Laplace equation. Theonly non-singular solution of this equation is a constantprofile, so that T (r, t) at r < R is reduced to T (t), wheretime t once again is regarded as a parameter. As for thetemperature field in the environmental fluid, it keeps the

same profile as that at R� δ, which eventually [bearingin mind Eq. (9)] brings about the following expressionfor Ts:

Ts =σI(t)

4πRκf≡ QabsRI(t)

4κf. (10)

Case L2: δ � √χfτ � R � √χpτ . In this casethe field inside the particle is still quasi-steady, so thatits r-dependence may be neglected, see above, case (L1).Regarding the field outside the particle, it is essentiallyt- and r-dependent. To determine Ts we may employthe energy conservation law. For simplicity we considera rectangular laser pulse with intensity I0 and durationτ . To a certain moment of time t ≤ τ the energy Wabsorbed by the particle is σabsI0t.

The absorbed energy is consumed to heat the particleto temperature Ts and to heat an adjacent layer of thefluid. The former requires the energy (4/3)πR3CpρpTs,the latter 4πR22

√χf tCfρfTs/2, (to enhance accuracy of

the estimate we have taken into account that the scaleof a layer heated to time t is 2

√χf t [7] and replaced the

profile of the temperature in the heated layer by its meanvalue Ts/2). Here C and ρ stand for the specific heat anddensity of the particle (subscript p) and fluid (subscriptf), respectively.

Equalizing W to the consumed energy and consideringthe equality as an equation for unknown Ts, one easilyderives

Ts(t) =σabsI0t

43πR

3Cpρp + 4πR2√χf tCfρf

≡ QabsI0t43RCpρp + 4

√χf tCfρf

. (11)

The obtained Ts(t) is a monotonic function of t, so themaximal temperature is achieved in the end of the laserpulse. Replacement t → τ brings about the correspond-ing expression for the maximal temperature Ts(max).

Case L3: δ � √χfτ �√χpτ � R. According to

the employed problem formulation this case correspondto heating of infinite compound space whose left semi-space has the thermometric properties of the particle,the right one those of the fluid, and energy is releasedat the boundary between the semi-spaces. This problemis exactly solvable [7]. The solution yields the followingexpression for the surface temperature:

Ts(t) =σI0

2R2π√π

√χpχf t

κp√χf

+ κf√χp

≡ αI02π√π

√χpχf t

κp√χf

+ κf√χp

(12)

Once again Ts occurs a monotonic function of time, sothe maximal temperature is achieved in the end of thepulse, at t = τ .

4

Case L4:√χfτ � δ � R � √χpτ . The surface

absorption of light. The case is identical to (L2), so thatEq. (11) is valid.

Case L5:√χfτ � δ � √χpτ � R. The case is

identical to (L3), and Eq. (12) is valid.

Other cases

Case O1:√χfτ � R � δ � √χpτ . From the view-

point of the heat transfer the case is equivalent to (S2),but the absorption cross-section is described by Eq. (1).As a result

Ts(t) =σabsI0t

43πR

3Cpρp + 4πR2Cfρf√χf t

≡ αkI0t

43πCpρp + 4πCfρf

√χf t

R

. (13)

Case O2:√χfτ � R � √χpτ � δ. The case is

identical to (O1), and Eq. (13) is valid.Case O3:

√χfτ �

√χpτ � R � δ. According to

the approximation used the energy is released inside theparticle in a spatially-uniform manner. The case occursanalogous to(O1), and again Eq. (13) is valid.

Case O4:√χfτ �

√χpτ � δ � R. In this case the

energy release occurs in a surface layer with the volume4πR2δ. The condition

√χpτ � δ allows to neglect dis-

tortion of the temperature inside the layer by heat con-ductivity. Thus, the released energy is consumed to heatthe mentioned layer and the layer of fluid with thickness2√χf t. The energy balance yields the following expres-

sion for Ts:

Ts(t) =σabsI0t

4πR2(Cpρpδ + Cfρf√χf t)

≡ QabsI0t

4(Cpρpδ + Cfρf√χf t)

. (14)

The maximal temperature is achieved in the end of thelaser pulse, at t = τ .

Thus the entire set of possible relation between thefour length scales of the problem has been inspected.

NUMERICAL VS. ANALYTICAL RESULTS

Let us discuss the results obtained. Note that the de-pendence of the dimensionless quantities α and Qabs on Rusually is very weak except for a narrow region centeredabout R ∼ δ, where the dependence σabs ∼ R3 is replacedby σabs ∼ R2. To illustrate this point the dependenciesσabs(R) and the corresponding Qabs(R) for a sphericalgold particle are presented in Fig. 1. The calculationsare made according to the exact Mie solution [8] and ac-tual optical constants of gold [9] at the wavelength of the

10-16

10-14

10-12

10-10

10-8

10-6

σab

s [cm

2 ]

10-7 10-6 10-5 10-4 10-3 10-2

radius, R [cm]

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

Qabs

σabs Qabs

FIG. 1. (color online) Absorption cross-section σabs and thecorresponding dimensionless efficiency Qabs for a sphericalgold particle at λ = 532 nm and δ(λ) = 22 nm as functionsof the particle radius R. Calculations according to the exactMie solution.

0.01

0.1

1

10

Tem

pera

ture

, T [

C]

10-7

10-6

10-5

10-4

10-3

10-2

radius, R [cm]

Numerical results Analitical results

FIG. 2. (color online) Comparison of the analytical and nu-merical results for the maximal temperature of a sphericalgold particle with radius R in water heated by a rectangularlaser pulse with wavelength 532 nm, I0 = 2 · 105 W/cm2 andτ = 50 ns. [δ � √χfτ , see cases (S1) and (L1)-(L3)].

incident light in a vacuum λ = 532 nm. The correspond-ing value of δ calculated as c/(ωn′′Au), where n′′Au standsfor the imaginary part of the refractive index of gold atthe given wavelength, is 22 nm. The cubic dependenceσabs(R) [linear dependence of Qabs, i.e. independence ofα of R] at R < δ is seen straightforwardly. At R > δthe efficiency Qabs drops from 0.6 to 0.2 when R varies

5

2

4

68

0.01

2

4

68

Tem

pera

ture

, T [

C]

10-7

10-6

10-5

10-4

10-3

10-2

radius, R [cm]

Numerical results Analytical results

FIG. 3. (color online) Same as in Fig. 2 at τ = 50 ps.[√χfτ � δ � √χpτ , see cases (S2), (O1), and (L4)].

10-5

2

3

4

567

10-4

Tem

pera

ture

, T [

C]

10-7

10-6

10-5

10-4

10-3

10-2

radius, R [cm]

Numerical results Analytical results

FIG. 4. (color online) The same as that shown in Fig. 2 atτ = 50 fs. [

√χpτ � δ, see cases (S3), and (O2)-(O4)].

in three order of magnitude, i.e. the dependence Qabs(R)is extremely weak, and our assumption σabs ∼ R2 doescapture the main R-dependence of σabs in this area.

It means that the main dependence of Ts(max) on theparticle size is given by the explicit R-dependence ofEqs. (8), (10)–(14). In particular, for cases (i)–(iv) thequadratic growth of Ts(max) with an increase in R is re-placed by the linear at R ∼ δ, then Ts(max) reaches itsmaximum at R ∼ √χfτ , declines with further increasein R and finally approaches to a constant at R� √χpτ .

τ

FIG. 5. (color online) Density plot of the maximum of thesurface temperature Ts(max) (arbitrary units) for a gold par-ticle in water irradiated by a rectangular laser pulse withI0 = 2 · 105 W/cm2 as a function of the duration of the pulseτ and the particle radius R. The dashed line correspondsto R = 2

√χfτ . Above this line the temperature becomes

τ -independent, see Eqs. (8), (10).

FIG. 6. (color online) The radius Rmax providing the absolutemaximum of the surface temperature and the correspondingtemperature Ts(Max) as functions of the laser pulse durationτ for a gold particle in water irradiated by a rectangular laserpulse with I0 = 2 · 105 W/cm2. The dashed line indicates de-pendence R = 2

√χfτ (a). The initial part of plot Ts(Max)(τ),

shown in panel (a), in Log-Log scale (b).

For other case Ts(max)(R) is treated in a similar manner.

Accuracy of the developed approach is illustrated bycomparison of the obtained analytical expressions withnumerical simulations of the corresponding spherically-symmetric version of the heat conduction equation for agold particle in water presented in Fig. 2–4, where σabsis taken from the exact Mie solution for the gold particle.

6

The particle is heated by a rectangular laser pulse, whoseintensity for definiteness is assigned the typical value I0 =2 · 105 W/cm2, and various values of τ (indicated in thefigure captions).

Note, that there are two competing mechanisms ofmaximization of the surface temperature. The first is re-lated to optics being associated with the local maximumof Qabs at R ' δ, see Fig. 1. The second is related to theheat transfer problem. It is associated with the changeof the quasi-steady, R-dependent temperature field in thevicinity of the particle to the essentially time-dependenttemperature profile, which does not depend on R. Thechange occurs at R ≈ 2

√χfτ .

To understand the relative role of these mechanisms weplot the Ts(max) as a function of R and τ , see Fig. 5. Atevery given R the Ts(max) increases monotonically withan increase in τ until the latter reaches the values τ 'R2/χf . Then, Ts(Max) becomes τ -independent.

Next, we perform the following calculations. For everygiven τ we find such a value of R = Rmax that maxi-mizes Ts(max)(R), i.e., provides the absolute maximumof the surface temperature, which may be achieved forthe given τ . Then, Rmax and the corresponding temper-ature Ts(Max) = Ts(max)(Rmax) are plotted as functionsof τ , see Fig. 6. It is seen straightforwardly that for theproblem in question heating for several degrees and highbegins from τ ≥ 10−9 s, when the heat transfer mecha-nism prevails over the optic one. Let us stress that, asit has been already pointed out in Sec. II, owing to thelinearity of the problem the temperature is just propor-tional to I0, which allows easily recalculate the resultsobtained for a given value of I0 to any other its value.

To illustrate how our results may be employed in vari-ous applications let us consider an important example ofselective laser photo-thermal therapy of cancer. Thus, itwas shown that 40 nm gold nanoparticle conjugated tocertain antibodies and then incubated with both humanoral cancer cells and nonmalignant skin cells were prefer-entially and specifically bound to the cancer cells. Next,the nanoparticle-labeled cells were exposed to a CW ar-gon ion laser at 514 nm. It was found that the malignantcells required less than half the laser energy to be killedas compared to the benign cells. The destruction of thecancer cells occurred owing to laser heating of the goldnanoparticles up to a certain threshold temperature [10].

On the other hand, applying this approach in vivo toavoid unwanted effects one should minimize the expo-sure to the laser beam of the benign tissues and even thetumor itself, maximizing the rate of energy delivered tothe nanoparticles and stored in the particles and theirimmediate vicinity. To this end a pulse laser should beemployed. Application of our results indicates that re-duction of the pulse duration from infinity (for a CWlaser) to a nunosecond scale (without change of I0) prac-tically does not affect the maximal temperature rise ofthe nanoparticles. Presumably, heating of the nanopar-

ticles by such a pulse still should be fatal for the cancercells bound to them, while the exposure of the rest of thetumor and the benign tissues will be reduced dramati-cally.

CONCLUSION

We have demonstrated that rather a complex problemof laser pulse heating of spherical metal particles embed-ded in a transparent fluid may be described by relativelysimple analytical expressions, which provide the depen-dence of the maximum temperature at the particle sur-face Ts(max) on the particle size and other parameters ofthe problem. We have demonstrated, by a direct com-parison with the numerical simulations, that these ex-pressions remain valid for any practically important sizeof the particle and duration of the laser pulse. More im-portantly, at fixed values of the material constants, thefunction Ts(max)(R) reaches local maxima at R ' δ andR ∼ √χfτ . For a gold particle in water considerableheating begins from τ ≥ 10−9 s, when the absolute max-imum of the temperature is achieved at R ≈ 2

√χfτ . We

believe these results not only give the most general so-lution of the problem but will also be useful to optimizethe efficiency of laser pulse heating of nanoparticles invarious problems of laser-matter interaction.

ACKNOWLEDGEMENTS

The work was partially supported by the AustralianResearch Council. M.T. thanks the Max-Planck-Institutfur Physik komplexer Systeme for kind hospitality duringthe substantial part of this project.

∗ E-mail: tribelsky˙at˙mirea.ru† E-mail: Boris˙L˙at˙dsi.a-star.edu.sg

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